Splitting Lemma

Splittin Lemma. Let be a short exact sequence of module homomorphism. The the following are equivalent.

(Right split): There is an module homomorphism with ,

(Left split): There is an module homomorphism with ,

(Direct Sum): The given sequence is isomorphic to , and thus .

Proof.
(1) (3). By the universal property of coproduct, we have a morphism with , and . Moreover, . Apply it by , and then we have . Since by exactness, then . Thus we have proved . Hence, they all commute between these morphisms, and is an isomorphism by the short five lemma.

(2) (3). By the duality of the above statement, we have the dual result. But we could also prove it directly.

By the universal property of product, we have with and . Moreover . Hence . Thus we have . It follows that is an isomorphism by the short five lemma.

(3) (1), (2). Let and . We have and .

Short Five Lemma 比 Five Lemma 簡單多了，用這個會比較好作。另外關於 split 還有

Lemma. If and , then .
Proof . Let and . Sinc . and . We have proved that .

類似的還有一個，這相當於線代中對投影矩陣的空間拆解。

Lemma. If for is a -homomorphism. Then
Proof. Let and . It’s obvious that , and which means . Hence .